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arXiv:2210.03738 · Physical Review Letters

Continuum of Bound States in a Non-Hermitian Model

Qiang Wang, Changyan Zhu, Xu Zheng, Haoran Xue, Baile Zhang, Y. D. Chong

Published Mar 8, 2023• 32 citations
arXivHTMLPDF

Abstract

In a Hermitian system, bound states must have quantized energies, whereas free states can form a continuum. We demonstrate how this principle fails for non-Hermitian systems, by analyzing non-Hermitian continuous Hamiltonians with an imaginary momentum and Landau-type vector potential. The eigenstates, which we call "continuum Landau modes" (CLMs), have Gaussian spatial envelopes and form a continuum filling the complex energy plane. We present experimentally realizable 1D and 2D lattice models that host CLMs; the lattice eigenstates are localized and have other features matching the continuous model. One of these lattices can serve as a rainbow trap, whereby the response to an excitation is concentrated at a position proportional to the frequency. Another lattice can act a wave funnel, concentrating an input excitation onto a boundary over a wide frequency bandwidth. Unlike recent funneling schemes based on the non-Hermitian skin effect, this requires a simple lattice design with reciprocal couplings.

DOI
10.1103/physrevlett.130.103602

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In a Hermitian system, bound states must have quantized energies, whereas free states can form a continuum. We demonstrate how this principle fails for non-Hermitian systems, by analyzing non-Hermitian continuous Hamiltonians with an imaginary momentum and Landau-type vector potential. The eigenstates, which we call "continuum Landau modes" (CLMs), have Gaussian spatial envelopes and form a continuum filling the complex energy plane. We present experimentally realizable 1D and 2D lattice models that host CLMs; the lattice eigenstates are localized and have other features matching the continuous model. One of these lattices can serve as a rainbow trap, whereby the response to an excitation is concentrated at a position proportional to the frequency. Another lattice can act a wave funnel, concentrating an input excitation onto a boundary over a wide frequency bandwidth. Unlike recent funneling schemes based on the non-Hermitian skin effect, this requires a simple lattice design with reciprocal couplings.
We study a one-dimensional system subjected to a linearly varying imaginary vector potential, which is described by the single-particle continuous Schr\"odinger equation and is analytically solved. The eigenenergy spectrum is found to be real under open boundary condition (OBC) but forms a parabola in the complex energy plane under periodic boundary condition (PBC). The eigenstates always exhibit a modulated Gaussian distribution and are all pinned on the same position, which is determined by the imaginary vector potential and boundary conditions. These behaviors are in sharp contrast to the non-Hermitian skin effect (NHSE) in systems with constant imaginary vector potential, where the eigenstates are exponentially distributed under OBC but become extended under PBC. We further demonstrate that even though the spectrum under PBC is an open curve, the Gaussian type of NHSE still has a topological origin and is characterized by a nonvanishing winding number in the PBC spectrum. The energies interior to the parabola can support localized edge states under semi-infinite boundary condition. The corresponding tight-binding lattice models also show similar properties, except that the PBC spectrum forms closed loops. Our work opens a door for the study of quantum systems with spatially varying imaginary vector potentials.
Abstract An extensive number of the eigenstates can become exponentially localized at one boundary of nonreciprocal non-Hermitian systems. This effect is known as the non-Hermitian skin effect and has been studied mostly in tight-binding lattices. To extend the skin effect to continues systems beyond 1D, we introduce a quadratic imaginary vector potential in the continuous two dimensional Schrödinger equation. We find that inseparable eigenfunctions for separable nonreciprocal Hamiltonians appear under infinite boundary conditions. Introducing boundaries destroy them and hence they can only be used as quasi-stationary states in practice. We show that all eigenstates can be clustered at the point where the imaginary vector potential is minimum in a confined system.
Non-Hermitian lattices can host the non-Hermitian skin effect, a boundary-induced collapse of all bulk eigenstates into exponentially localized edge modes. This effect underlies anomalous bulk-boundary correspondence and remarkable enhancements in non-Hermitian sensing, yet direct energy-resolved access to the eigenmodes of non-Hermitian lattices has remained limited. Here we report band- and energy-resolved eigenmode spectroscopy of skin modes in a frequency synthetic dimension. By introducing strong frequency-domain boundaries in an electro-optically modulated ring resonator, we realize finite non-Hermitian lattices and use laser detuning as a spectroscopic axis for the eigenenergies of the effective Hamiltonian. Site-resolved heterodyne measurements then reconstruct the spatial profile of each mode, revealing boundary-localized skin states throughout the spectrum and their eigenenergy-dependent displacement from the edge. Beyond 1D, the same frequency-boundary architecture, upon incorporating long-range couplings between finite lattices, produces genuine 2D frequency lattices rather than the hitherto-realized folded 1D systems on twisted tubes. In these lattices we observe tunable directional transport and edge localization in two synthetic dimensions. Our results introduce eigenmode spectroscopy as a direct probe of non-Hermitian physics and establish strongly bounded frequency lattices as a flexible platform for Hamiltonian engineering.
In this paper, we investigate the non-Hermitian skin effect in a semi-infinite Fock-state lattice, where the inherent coupling scales as \sqrt{n}. By analytically solving a non-uniform, non-reciprocal SSH model, we demonstrate that the intrinsic inhomogeneous coupling, in combination with nonreciprocity, fundamentally modifies the conventional skin effect. Instead of accumulating at the physical boundary, all eigenmodes become compressed and skewed within a finite spatial range determined by the inhomogeneous profile-a phenomenon we term the confined non-Hermitian skin effect. Consequently, the evolution of the probability distribution on the lattice starting from a single site is doubly confined: it is spatially bounded to a finite range by the inhomogeneous coupling, and further restricted to a one-sided trajectory at the edge of this range by the non-reciprocity. Moreover, a feasible experimental scheme based on a single trapped ion is also proposed. This work reveals how engineered coupling profiles in synthetic dimensions can reshape non-Hermitian properties and enable new protocols for quantum state manipulation.
We study the one-dimensional non-Hermitian lattices with linearly varying nonreciprocal hopping, where the non-Hermitian skin effect (NHSE) is found to be dissolved gradually as the strength of nonreciprocity increases. The energy spectrum under the open boundary condition is composed of real and imaginary eigenenergies when the nonreciprocal hopping is weak. Interestingly, the real eigenenergies form an equally spaced ladder, and the corresponding eigenstates are localized at the boundary with a Gaussian distribution due to NHSE. By increasing the nonreciprocity, the number of real eigenenergies will decrease while more and more eigenenergies become imaginary. Accompanied by the real-imaginary transition in the spectrum, the eigenstates are shifted from the boundary into the bulk of the lattice. When the nonreciprocity gets strong enough, the whole spectrum will be imaginary and the NHSE disappears completely in the system; i.e., all the eigenstates become Gaussian bound states localized inside the bulk. Our work unveils the exotic properties of non-Hermitian systems with spatially varying nonreciprocal hopping.
The non-Hermitian skin effect and nonreciprocal behavior are sensitive to the boundary conditions, which are unique features of non-Hermitian systems. In such systems, eigenenergies can become complex, and all eigenstates tend to localize at the boundary, a phenomenon that contrasts with Hermitian topologies. In this work, we theoretically study the dynamic behavior of the propagation of Gaussian wavepackets inside a non-Hermitian lattice and analyze the self-acceleration process of bulk state or Gaussian wavepackets toward the system's boundary. The initial wavepackets will not only propagate toward the side where the eigenstates are localized, but also their momentum will approach to a specific value. This value corresponds to the maximum imaginary components of the energy dispersion. In addition, if the wavepackets in the momentum space cover this specific momentum, they will eventually exhibit exponentially increasing amplitudes with time evolution, maintaining the dynamic protected condition for an extended period of time until they approach the boundary. We also take two widely used toy models as examples in one and two dimensions to verify the correspondence of the non-Hermitian skin effect and the dynamic protected state.